metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic24, C16.S3, C3⋊1Q32, C6.3D8, C48.1C2, C4.3D12, C2.5D24, C8.15D6, C12.26D4, C24.16C22, Dic12.1C2, SmallGroup(96,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic24
G = < a,b | a48=1, b2=a24, bab-1=a-1 >
Character table of Dic24
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 48A | 48B | 48C | 48D | 48E | 48F | 48G | 48H | |
| size | 1 | 1 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ11 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ12 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | -√3 | √3 | √3 | -√3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | orthogonal lifted from D24 |
| ρ13 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | √3 | -√3 | -√3 | √3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | orthogonal lifted from D24 |
| ρ14 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | -√3 | √3 | √3 | -√3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | orthogonal lifted from D24 |
| ρ15 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | √3 | -√3 | -√3 | √3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | orthogonal lifted from D24 |
| ρ16 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | -√2 | -√2 | √2 | √2 | ζ1615-ζ169 | ζ165-ζ163 | ζ165-ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | √2 | √2 | -√2 | -√2 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ18 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | ζ165-ζ163 | -ζ1615+ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -√2 | -√2 | √2 | √2 | -ζ1615+ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | ζ165-ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | symplectic lifted from Q32, Schur index 2 |
| ρ19 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | √2 | √2 | -√2 | -√2 | -ζ165+ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ165-ζ163 | symplectic lifted from Q32, Schur index 2 |
| ρ20 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ167+ζ16 | -ζ165+ζ163 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ167ζ32+ζ16ζ32+ζ16 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ165ζ32+ζ165+ζ163ζ32 | ζ167ζ3+ζ16ζ3+ζ16 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ165ζ3+ζ163ζ3+ζ163 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | symplectic faithful, Schur index 2 |
| ρ21 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ167+ζ16 | -ζ165+ζ163 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ167ζ3+ζ16ζ3+ζ16 | ζ165ζ32+ζ165+ζ163ζ32 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ3+ζ163ζ3+ζ163 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | symplectic faithful, Schur index 2 |
| ρ22 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | √3 | -√3 | -ζ165+ζ163 | -ζ167+ζ16 | -ζ1615+ζ169 | ζ165-ζ163 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ167ζ32+ζ167+ζ16ζ32 | ζ165ζ3+ζ163ζ3+ζ163 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ167ζ3+ζ167+ζ16ζ3 | ζ165ζ32+ζ165+ζ163ζ32 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | symplectic faithful, Schur index 2 |
| ρ23 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | -√3 | √3 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ165ζ32+ζ165+ζ163ζ32 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ165ζ3+ζ163ζ3+ζ163 | symplectic faithful, Schur index 2 |
| ρ24 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | √3 | -√3 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | ζ165ζ32+ζ165+ζ163ζ32 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | ζ165ζ3+ζ163ζ3+ζ163 | ζ1613ζ3+ζ1613+ζ1611ζ3 | symplectic faithful, Schur index 2 |
| ρ25 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | √3 | -√3 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | ζ165ζ3+ζ163ζ3+ζ163 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | ζ165ζ32+ζ165+ζ163ζ32 | ζ1613ζ32+ζ1611ζ32+ζ1611 | symplectic faithful, Schur index 2 |
| ρ26 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | -√3 | √3 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ165ζ3+ζ163ζ3+ζ163 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ165ζ32+ζ165+ζ163ζ32 | symplectic faithful, Schur index 2 |
| ρ27 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | -√3 | √3 | -ζ165+ζ163 | -ζ167+ζ16 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ167ζ3+ζ167+ζ16ζ3 | ζ1613ζ3+ζ1613+ζ1611ζ3 | ζ165ζ3+ζ163ζ3+ζ163 | ζ167ζ32+ζ167+ζ16ζ32 | ζ1613ζ32+ζ1611ζ32+ζ1611 | ζ165ζ32+ζ165+ζ163ζ32 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | symplectic faithful, Schur index 2 |
►Regular action on 96 points
Generators in S96
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 72 25 96)(2 71 26 95)(3 70 27 94)(4 69 28 93)(5 68 29 92)(6 67 30 91)(7 66 31 90)(8 65 32 89)(9 64 33 88)(10 63 34 87)(11 62 35 86)(12 61 36 85)(13 60 37 84)(14 59 38 83)(15 58 39 82)(16 57 40 81)(17 56 41 80)(18 55 42 79)(19 54 43 78)(20 53 44 77)(21 52 45 76)(22 51 46 75)(23 50 47 74)(24 49 48 73)
G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,72,25,96)(2,71,26,95)(3,70,27,94)(4,69,28,93)(5,68,29,92)(6,67,30,91)(7,66,31,90)(8,65,32,89)(9,64,33,88)(10,63,34,87)(11,62,35,86)(12,61,36,85)(13,60,37,84)(14,59,38,83)(15,58,39,82)(16,57,40,81)(17,56,41,80)(18,55,42,79)(19,54,43,78)(20,53,44,77)(21,52,45,76)(22,51,46,75)(23,50,47,74)(24,49,48,73)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,72,25,96)(2,71,26,95)(3,70,27,94)(4,69,28,93)(5,68,29,92)(6,67,30,91)(7,66,31,90)(8,65,32,89)(9,64,33,88)(10,63,34,87)(11,62,35,86)(12,61,36,85)(13,60,37,84)(14,59,38,83)(15,58,39,82)(16,57,40,81)(17,56,41,80)(18,55,42,79)(19,54,43,78)(20,53,44,77)(21,52,45,76)(22,51,46,75)(23,50,47,74)(24,49,48,73) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,72,25,96),(2,71,26,95),(3,70,27,94),(4,69,28,93),(5,68,29,92),(6,67,30,91),(7,66,31,90),(8,65,32,89),(9,64,33,88),(10,63,34,87),(11,62,35,86),(12,61,36,85),(13,60,37,84),(14,59,38,83),(15,58,39,82),(16,57,40,81),(17,56,41,80),(18,55,42,79),(19,54,43,78),(20,53,44,77),(21,52,45,76),(22,51,46,75),(23,50,47,74),(24,49,48,73)]])
Dic24 is a maximal subgroup of
C32⋊S3 Dic48 D16.S3 C3⋊Q64 D48⋊7C2 C16.D6 D16⋊3S3 SD32⋊S3 S3×Q32 Dic72 C32⋊3Q32 C32⋊5Q32 C5⋊Dic24 Dic120
Dic24 is a maximal quotient of
C2.Dic24 C48⋊5C4 Dic72 C32⋊3Q32 C32⋊5Q32 C5⋊Dic24 Dic120
Matrix representation of Dic24 ►in GL2(𝔽47) generated by
| 32 | 11 |
| 11 | 42 |
| 0 | 46 |
| 1 | 0 |
G:=sub<GL(2,GF(47))| [32,11,11,42],[0,1,46,0] >;
Dic24 in GAP, Magma, Sage, TeX
{\rm Dic}_{24} % in TeX
G:=Group("Dic24"); // GroupNames label
G:=SmallGroup(96,8);
// by ID
G=gap.SmallGroup(96,8);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,79,218,122,579,69,2309]);
// Polycyclic
G:=Group<a,b|a^48=1,b^2=a^24,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic24 in TeX
Character table of Dic24 in TeX